## Saturday, October 31, 2015

### Impulse, Change of Momentum, Applications and Problems with Solutions

Change of momentum and its applications

Momentum is defined as a product of mass and velocity. It is a physical quantity which explains that how much kinetic energy is transferred between the two bodies during the collision. If a body has higher momentum, it transfers more kinetic energy to the other bodies and vice versa. Momentum is a vector quantity which has both magnitude and direction. When we are calculating the change in the momentum of a body in between any two cases, we shall use parallelogram law of vectors.

If a body is coming back with the same velocity after striking an obstacle vertically, the momentum of the body reminds the same in terms of the magnitude but the direction is opposite. Hence the difference in the magnitude of the momentum is equal to double the initial momentum of the system. It is simply because we need to calculate the difference and hence we need to use a negative sign. We also need to use another negative sign because the final momentum is opposite to the initial momentum.

Application One

Let us consider a ball of mass M moving with the velocity striking of all making an angle with the horizontal. Let after striking the obstacle it bounce back with the same velocity and again by making the same angle with the horizontal. In this case we can calculate the change in the momentum as shown below.

Application Two

Let us consider another situation where gravel is dropped vertically at at a constant rate on the conveyor belt which is moving horizontally. As this gravel is acting agonist the motion of the horizontal system, we need to apply a constant force to keep it moving. We can get the value of that force simply by equating the force to the definition of rate of change of momentum.

Application Three

Let us consider another case where water is coming out of the nozzle through a horizontal hosepipe of a certain area of cross-section. If the water is coming with a constant speed, in the horizontal force on the pipe to prevent it to recoil can be calculated using the definition of the force in terms of rate of change of momentum as shown below.

Laugh conservation of linear momentum

If there is no external force acting on a system, the linear momentum of the system always remains constant. This is called conservation of linear momentum are simply momentum. Linear momentum is conserved in all the cases if there are no external forces acting on the system. This is one of the fundamental conserving physical quantities in mechanics. Whatever may be the kind of the collision between the two bodies, linear momentum is always conserved.

Impulse and its applications

Impulse is a physical quantity which is defined as the product of force and time. It is actually a physical quantity which explains how a large forces acting on a body for a very short interval of time. This is in terms of magnitude is equal to the linear momentum.

Example one

We can see the examples of the impulsive force acting on a body in daily life at many situations. When the cricket ball is hit by a bat, the batsmen hits the ball with a large force in a very short interval of the time and hence a large impulsive forces acts on the ball. We can see a small deformation in the shape of the ball also when the ball is little bit weak to absorb as much of the force.

Example Two

When a fielder in the cricket match tries to catch a ball, he pulls his hands back word in the process. This is a common observation that any people who are familiar with the cricket know it and see it. If he didn’t pull his hands backward, the ball hits his palms with a larger force in a very short interval of time and it causes injury to his palms. To avoid this he pulls the hands in the backward direction and hence he is not allowing the entire large force acting on the body in a very short interval of time, rather is giving the extra time for the ball to come to the state of rest. In this process the force acting on the palm is distributed over the time and per second that much of the large force is not acting on the palm and hence he is not going to be injured.

Example Three

If two people are jumping from the same height onto the ground, the person jumping onto the cement floor get more injured than the person who is jumping from the same height but onto the sand floor. The logic is simple. The person jumping onto the cement floor comes to the state of rest immediately and hence the large impulsive force acts on his body which causes injury. The person jumping onto the sand gets extra time as the penetration little bit deep into the sand and hence he has less possibility of getting that much injury.

Example Four

We also know that the shock absorbers are used in all the motor vehicles to make the journey comfortable. The shock absorbers do the same job and get more time for the force distribution on the bodies of the people sitting in the vehicle and hence less impulsive force will be acting on them. So the journey will be more comfortable.

Problem

A machine gun has a mass of 20 kg. It fires bullet of mass 35 gram at the rate of four bullets per second. The bullet speed is 400 meter per second. What must be the force that has to be applied to the gun to keep it on its position?

Solution

As there is no external force acting on the system, according to laugh conservation of linear momentum initial momentum of the system is equal to the final momentum. Therefore the momentum of the gun is equal to the sum of the momentum of the bullets. By equating that we can calculate the velocity of the recoil of the gun. We can also equal to the momentum in terms of the magnitude to a physical quantity called impulse.
By equating the momentum to the force we can solve the problem as shown below.

Problem

A disc of March 0.1 kg is floating horizontally in midair by firing the bullets of mass 0.05 kg each vertically at it at the rate of 10 bullets per second. If the bullets rebound with the same speed what is the speed with which each bullet has to be fired?

Solution

For the disc to be in the state of equilibrium, the resultant force acting on it shall be equal to 0. The force acting in the downward direction is the weight of the disc and the force acting in the upward direction is due to the change in the momentum. If these two are equal in magnitude as they were already in the opposite direction, the disc can be in the state of equilibrium. Applying this concept we can solve the problem as shown below.

Problem

The linear momentum of a particle is varying with the time as shown below. What is the relation between the force and time in this case?

Solution

We can define the force as rate of change of momentum. By differentiating the given momentum once with respect to the time hence we can get the force. Differentiation is a mathematical tool to get the microscopic physical quantity from the macroscopic physical quantity. By doing so we can get the relation and we can also draw a graph is as shown below.

Problem

A liquid of known density is passing through a horizontal pipe of uniform area of cross-section. The pipe is a right angled and the liquid is travelling with a particular velocity. What force has to be exerted at the bend to hold the pipe in the equilibrium position?

Solution

We can calculate the change of the momentum at the right angle using the parallelogram law of vectors. We can also equate the change in the momentum to the force basing on the Newton’s second law. By doing so, we can get the force as shown below.

Problem

A body of mass 10 kg is acted upon by force which is varying with respect to the time as shown below. The initial speed of the body is 10 m/s. What is the velocity of the body after five seconds?

Solution

Until now we are dealing with a constant force. But in this problem force is not constant but is varying with respect to time. To solve this kind of the problems, we need to use a mathematical tool called integration. In that case we can get the value of the macro picture by knowing the value of the micro picture. It is performed as shown in the below diagram.

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